How to show 111.........1(91 times) is a prime number?

Question

Is there any hint to find a given number is prime?

Answer 1

In general, and this'd be interesting for you to prove it, if you have 11111....1 ( n 1's), and we can write n = pm, all integers, then 11111....1 (n 1's) is divisible by 111....1 (p 1's), and in fact we get:
11....1 (n 1's) = (11...1)*(100..0100...01......1), where the right factor in the right side is composed of 1, then p-1 0's, then 1, then p-1 0's, etc., up to the last 1 which is the m-th one.
For example, with n = 15 and p = 5:
111111111111111 = (11111)(10000100001) , as you can easily check.
So now, since 91 = 7*13, we can write
11...1(91 1's) = (1111111)*(100000010000001000000100000100000010000001000000100000010000001000000100000010000001) -- 13 1's in the right, with 6 0's between them)

Regards
Tonio

Answer 2

Since 1111...1(91 times) is not prime,
as has been shown, let's answer the second
question:
One way to show a number is not prime
is to try to use the contrapositive of Fermat's
little theorem. Let (a,n) = 1.
If a^(n-1) != 1(mod n) then n is composite.
However, if a^(n-1) does equal 1 mod n, n
may still be composite.
Then n is called a pseudoprime base a.
For example, 2^340 = 1(mod 341)
but 341 is divisible by 11.
So you may ask, why not change bases and try
again?
For example 3^340 = 56(mod 341)
so now we know that 341 is composite.
BUT
There are infinitely many composite n, for which
the test fails for every base. These are called
Carmichael numbers. The smallest of
these is 561.
This idea, though, leads to probabilistic
prime testing: If we pick a hundred bases a prime to n.
and get a^(n-1) = 1(mod n) for
each such a, then n is most probably prime.
That's how the probable primes in the RSA
encryption code are chosen.
There are other tests of this sort, which
you can find in Knuth's ART OF COMPUTER
PROGRAMMING, vol.2.
Primality testing is a fascinating subject
which leads to a lot of beautiful mathematics.

Answer 3

the above is not a prime as 1 repeated 91 times is divsible by 1 repeated 7 times and also 1 repeated 13 times. 91 = 7 * 13

Answer 4

A number is prime if its divisible only by 1 and itself.

this is the only hint to find if a number is a prime or not.

Answer 5

CSI: CSI: NY CSI: Miami The Office 30 Rock House M.D. Lie to Me I'M GOING TO DIE OVER THE SUMMER! I CAN'T LIVE WITHOUT ANY OF THE SHOWS! I got mad even when ALL the CSI:'s took a break for a week I HAVEN'T MISSED ANY OF THE FINALES AND NEVER WILL! THEY ARE AWESOME SHOWS! AND IT IS 11:27 AM WHERE I AM AND I AM TIRED AND JUST SAW STAR TREK AND AM ACTING LIKE A NERD BUT ALSO AM ONLY LIKE 13 BUT SERIOUSLY LIKE EXTREMELY OBSESSED WITH ALL THE SHOWS I TYPED AND NOW I CANT STOP TYPING IN CAPS AND NOW IM LIKE FORGETTING WHAT IM GONNA SAY AND WHEN I TYPE OBSESSED I ALMOST TYPED OBESESSED IS THAT EVEN A WORD!!!?!?!?!? THE END

Answer 6

91 has 7 and 13 as prime factors. 91 is therefore not prime

Meaning 11111....(91 times) cannot be prime as it has 1111111 and 11111... (13 times) as prime factors.

Answer 7

try using mathematical induction and then contradiction.